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What are natural numbers? - Mathematics Stack Exchange
What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational numb...
Is $0$ a natural number? - Mathematics Stack Exchange
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered i...
What is a natural number? - Mathematics Stack Exchange
This set N N is a subset of the real numbers, and behaves (in every meaningful way) as the natural numbers— N N is isomorphic to N N. So perhaps it is reasonable to call N N the set of real-natural numbers.
I don’t know what a natural number actually is
The usage of the natural numbers and all the properties you know and love came first. The definition involving set theory and emptysets and such came after and was specifically constructed in such a way so as to make sure that natural numbers had all the properties we wanted and were used to, but so that it was consistent and able to be unambiguously defined "from the ground up" using set ...
discrete mathematics - What is the difference between natural numbers ...
Alternatively, if you were asked to define the natural numbers, you could do so axiomatically. With this method, you begin by declaring an element is a natural number, and it really doesn't matter what this element is called, be it 0 0 or 1 1 or Δ Δ - this is the idea I'm trying to convey.
Sum of all natural numbers. - Mathematics Stack Exchange
Well, as I said before, string theorists need to use the sum of all natural numbers and if they use the value−1 12 1 12 they get accurate results (is what I have heard).
Why set of natural numbers is infinite, while each natural number is ...
You can't seriously believe there are only finitely many numbers. "If a natural is finite, then that natural plus one is finite, hence the set of all naturals is finite" - this is a non sequitor, the set of all naturals is not itself a natural number.
Cardinality natural numbers - Mathematics Stack Exchange
N + 1 N + 1 is not a cardinal. Cardinal Number (wikipedia page) can read for more detailed information. If we look natural numbers as a set, we write N N. If we look them as an ordinal, we write ω ω, and for cardinal, we write ℵ0 ℵ 0. Then the answer is very easy, since no ordinal is an element of itself.
Why is $\\mathbb{Z}^{+}$ sometimes used to denote the natural numbers?
N N The set of natural numbers, including 0 [31], though sometimes also used to mean the same thing but excluding 0. [15] The " [31]" is a citation for Princeton University Press book, the " [15]" is for a Springer-Verlag book. Both books were published less than a decade ago.
The set of all finite subsets of the natural numbers is countable
Then is there a problem with my definition? I've thought I'd defined the set X as the set of all finite subsets of the natural numbers.
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